Probability Generating Functions Suppose a RV X can take on values in the set of non-negative integers: {0, 1, 2, …}. Definition: The probability generating.

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Presentation transcript:

Probability Generating Functions Suppose a RV X can take on values in the set of non-negative integers: {0, 1, 2, …}. Definition: The probability generating function of X is

Probability Generating Functions In general, the p.g.f. of the sum of independent RVs is the product of the p.g.f.s.

Geometric Distribution The waiting time for the first success in a sequence of Bernoulli trials (with success probability p) has a geometric distribution on {1, 2, …} with parameter p. Let T = the waiting time P(T = i) = q i  1 p (i = 1, 2, …)

Negative Binomial Distribution The waiting time for the r th success in a sequence of Bernoulli trials (with success probability p) has a negative binomial distribution on {r, r+1, r+2, …} with parameters r and p. Let T r = the waiting time for r th success

Exercise If a child exposed to a contagious disease has a 40% chance of catching it, what is the probability that the tenth child exposed is the third to catch the disease?

Poisson Distribution As we saw in Section 2.4, the Poisson distribution is a good approximation to the binomial distribution when n is large and p is small. The Poisson distribution has other interesting applications. N has a Poisson distribution with parameter  if